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Inequalities for Maximal Operators Associated with a Family of General Sets

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Abstract

Let \({\mathbb {E}}=\{E_r(x):r>0,x\in X\}\) be a family of open subsets of a topological space X equipped with a nonnegative Borel measure \(\mu \) satisfying some basic properties. We establish sharp quantitative weighted norm inequalities for the Hardy–Littlewood maximal operator \(M_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) in terms of mixed \(A_p\)–\(A_\infty \) constants. The main ingredient to prove this result is a sharp form of a weak reverse Hölder inequality for the \(A_{\infty ,{\mathbb {E}}}\) weights. As an application of this inequality, we also provide a quantitative version of the open property for \(A_{p,{\mathbb {E}}}\) weights. Next, we prove a covering lemma in this setting and using this lemma establish the endpoint Fefferman–Stein weighted inequality for the maximal operator \(M_{{\mathbb {E}}}\). Moreover, vector-valued extensions for maximal inequalities are also obtained in this context.

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References

  1. Aimar, H., Bernardis, A., Nowak, L.: Dyadic Fefferman–Stein inequalities and the equivalence of Haar bases on weighted Lebesgue spaces. Proc. R. Soc. Edinb. Sect. A 141, 1–21 (2011)

    Article  MathSciNet  Google Scholar 

  2. Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340, 253–272 (1993)

    Article  MathSciNet  Google Scholar 

  3. Caffarelli, L.A., Gutiérrez, C.E.: Real analysis related to the Monge–Ampère equation. Trans. Am. Math. Soc. 348, 1075–1092 (1996)

    Article  Google Scholar 

  4. Calderón, A.P.: Inequalities for the maximal function relative to a metric. Studia Math. 57, 297–306 (1976)

    Article  MathSciNet  Google Scholar 

  5. Calderón, A.P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math. 16, 1–63 (1975)

    Article  MathSciNet  Google Scholar 

  6. Carbery, A., Christ, M., Vance, J., Wainger, S., Watson, D.: Operators associated to flat plane curves: \(L^p\) estimates via dilation methods. Duke Math. J. 59, 675–700 (1989)

    Article  MathSciNet  Google Scholar 

  7. Chang, D.C., Dafni, G., Stein, E.M.: Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in \(\mathbb{R} ^d\). Trans. Am. Math. Soc. 351, 1605–1661 (1999)

    Article  Google Scholar 

  8. Chang, D.C., Krantz, S.G., Stein, E.M.: \(H^p\) Theory on a smooth domain in \(\mathbb{R} ^d\) and elliptic boundary value problems. J. Funct. Anal. 114, 286–347 (1993)

    Article  MathSciNet  Google Scholar 

  9. Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51, 241–250 (1974)

    Article  MathSciNet  Google Scholar 

  10. Ding, Y., Lee, M-Y., and Lin, C-Ch., \({{\cal{A}}}_{p,\mathbb{E}}\) weights, maximal operators, and Hardy spaces associated with a family of general sets. J. Fourier Anal. Appl., 20, 608–667 (2014)

  11. Ding, Y., Lee, M-Y., Lin, C-Ch.: Carleson measure characterization of weighted BMO associated with a family of general sets. J. Geom. Anal., 27, 842–867 (2017)

  12. Edwards, R.E., Gaudry, G.I.: Littlewood–Paley and Multiplier Theory. Springer-Verlag, Berlin (1977)

    Book  Google Scholar 

  13. Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)

    Article  MathSciNet  Google Scholar 

  14. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

    Google Scholar 

  15. Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Jpn., 22, 529–534 (1977/78)

  16. Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. North Holland Math. Studies, 116, North Holland, Amsterdam (1985)

  17. Guzmán, M.: Differentiation of integrals in \(\mathbb{R} ^d\). Lecture Notes in Mathematics, vol. 481. Springer, Berlin (1975)

  18. Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54, 81–116 (1930)

    Article  MathSciNet  Google Scholar 

  19. Hruščev, S,V.: A description of weights satisfying the \(A_\infty \) condition of Muckenhoupt. Proc. Am. Math. Soc., 90, 253–257 (1984)

  20. Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_{\infty }\) weights on spaces of homogeneous type. J. Funct. Anal. 263, 3883–3899 (2012)

    Article  MathSciNet  Google Scholar 

  21. Kenig, C.: Elliptic boundary value problems on Lipschitz domains, Bei**g Lectures in Harmonic Analysis. Ann. Math. Stud. 112, 131–183 (1986)

    Google Scholar 

  22. Kenig, C.: Harmonic analysis techniques for second order elliptic boundary value problems, vol. 83, CBMS Regional Conference Series in Math. AMS (1994)

  23. Kurtz, D.: Weighted norm inequalities for the Hardy–Littlewood maximal function for one parameter rectangles. Studia Math. 53, 39–54 (1975)

    Article  MathSciNet  Google Scholar 

  24. Luque, T., Parissis, I.: The endpoint Fefferman–Stein inequality for the strong maximal function. J. Funct. Anal. 266, 199–212 (2014)

    Article  MathSciNet  Google Scholar 

  25. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  Google Scholar 

  26. Neugebauer, C.: On the Hardy-Littlewood maximal function and some applications. Trans. Am. Math. Soc. 259, 99–105 (1980)

    Article  MathSciNet  Google Scholar 

  27. Ombrosi, S., Rivera-Ríos, I.P., Safe, M.D.: Fefferman–Stein inequalities for the Hardy–Littlewood maximal function on the infinite rooted k-ary tree. Int. Math. Res. Not., IMRN 2021, 2736–2762 (2021)

  28. Paternostro, V., Rela, E.: Improved Buckley’s theorem on LCA groups. Pac. J. Math. 299, 171–189 (2019)

    Article  Google Scholar 

  29. Saks, S.: Remark on the differentiability of the Lebesgue indefinite integral. Fund. Math. 22, 257–261 (1934)

    Article  Google Scholar 

  30. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    Google Scholar 

  31. Wiener, N.: The ergodic theorem. Duke Math. J. 5, 1–18 (1939)

    Article  MathSciNet  Google Scholar 

  32. Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic \(A_{\infty }\). Duke Math. J. 55, 19–50 (1987)

    Article  MathSciNet  Google Scholar 

  33. Wilson, J.M.: Weighted Littlewood–Paley theory and exponential-square integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)

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  1. Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Biswaranjan Behera & Md. Nurul Molla

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  1. Biswaranjan Behera
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  2. Md. Nurul Molla
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Correspondence to Md. Nurul Molla.

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Behera, B., Molla, M.N. Inequalities for Maximal Operators Associated with a Family of General Sets. Results Math 79, 197 (2024). https://doi.org/10.1007/s00025-024-02224-1

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